Adjoint Bernoulli’s Polynomials Coupling Gamma Functions and their Approximations

1. Introduction

Approximation theory serves as a fundamental pillar across numerous disciplines, offering a powerful framework for transforming intricate functions into simpler, more manageable forms. Its applications span mathematics, engineering, computational science, data analysis, and computer graphics. Within the realm of numerical analysis, approximation theory provides the foundational principles for developing efficient algorithms to solve complex mathematical problems. Its applications in applied mathematics are particularly significant in control theory, where the analysis of parametric curves and surfaces via control points and nets is essential for the design of control systems in diverse engineering contexts (Khan and Lobiyal, 2017; Khan et al., 2019). By enabling the simplification of complex curves and surfaces into fundamental mathematical forms, this theory also plays a crucial role in enhancing techniques for image rendering and symbolic computation. Furthermore, its broad utility has generated significant interest in fields such as medical science and other related disciplines.

The origins of “approximation theory can be traced back to 1715 when the English mathematician Taylor introduced the Taylor series, a summation of differentiable functions, to approximate diverse function classes. However, Taylor’s methodology was limited to finitely or infinitely differentiable functions. To overcome this limitation, The Weierstrass approximation theorem was introduced by Weierstrass (1985) in 1885. According to this theorem, it is possible to estimate any continuous function that can be defined on a closed interval with polynomial functions and with a precision that is uniform and as accurate as is needed. This groundbreaking theorem laid the foundation for numerous branches within approximation theory, such as numerical analysis, operator theory, and wavelet analysis. Despite its significance, the theorem’s initial proof was complex. Bernstein (1913) simplified” the proof using the binomial probability distribution:

where $p_{r\mathrm{,}k}\left(u\right)=\left(\begin{array}{c}r\\ k\end{array}\right)r^k{(1-r)}^{r-k}$. Bernstein established that $B_r(E;\cdot )\rightrightarrows E$ holds for every bounded function $E$ on [0,1], with $\rightrightarrows$ indicating uniform convergence.

Over recent decades, numerous modifications of the operators in Eq. (1) have been developed to enhance their approximation properties. These refinements have been studied across bounded and unbounded intervals within various functional spaces (Savas and Patterson, 2004; Ansari and Özger, 2024; Aslan, 2025; Sava and Mursaleen, 2023; Alamer and Nasiruzzaman, 2025; Özger et al., 2025; Mohiuddine et al., 2024).

2. Uniform rate of convergence and order of approximation

Definition 2.1 (Devore and Lorentz, 1993) For $g\in C_B\left[\mathrm{0,}\infty \right)$, we have:

and

where $\omega$ is the modulus of continuity (Aslan, 2025; Mansoori et al., 2025).

Theorem 2.1 Let ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ be given in Eq. (4), a sequence of operators and for $g\in C_B\left[\mathrm{0,}\infty \right)$. Then, ${\tilde{G}}_{r\mathrm{,}\lambda }(g;.)$ converges uniformly on closed interval $\left[\mathrm{0,}b\right]$, where $b<\infty )$.

Proof. It is adequate to prove (using Korovkin theorem (Altomare and Campiti, 1994)) that

${\tilde{G}}_{r\mathrm{,}\lambda }({\theta }^j;\mu )={\mu }^j\mathrm{,}j\in \{\mathrm{0,...,2}\},$

uniformly on each closed and bounded subset of $[\mathrm{0,}\infty ).$ Using Lemma (1.2), we arrive at the desired result immediately.

Next result is the study of order of approximation of Eq. (4) in terms of modulus of continuity in Eq. (8) as:

Theorem 2.2 Let $g\in C_B\left[\mathrm{0,}\infty \right)$. Then, the operators’ sequence ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ in Eq. (4), one has

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le 2\omega (g;\tilde{\delta }),$

where $\tilde{\delta }=\sqrt{{\tilde{G}}_{r\mathrm{,}\lambda }(g_2^{\mu };\mu )}$.

Proof. With the aid of definition of Eq. (4), we get

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|=\left|\frac{e^{-r\mu }}{e-1}\sum _{\nu =0}^{\infty }\frac{{\beta }_{\nu }\left(r\mu \right)}{\nu !}{\int }_0^{\infty }{b}_{r\mathrm{,}\nu }^{\lambda }\right(\theta \left)\right\{g\left(\theta \right)-g\left(\mu \right)\}d\theta |,$

$\le \frac{e^{-r\mu }}{e-1}\sum _{\nu =0}^{\infty }\frac{{\beta }_{\nu }\left(r\mu \right)}{\nu !}{\int }_0^{\infty }{b}_{r\mathrm{,}\nu }^{\lambda }\left(\theta \right)\left|g\right(\theta )-g(\mu \left)\right|d\theta$

$\le \frac{e^{-r\mu }}{e-1}\sum _{\nu =0}^{\infty }\frac{{\beta }_{\nu }\left(r\mu \right)}{\nu !}{\int }_0^{\infty }{b}_{r\mathrm{,}\nu }^{\lambda }\left(\theta \right)\left(1+\frac{|\theta -\mu |}{\tilde{\delta }}\right)\omega (g;\tilde{\delta })d\theta$

$\le \left(1+\frac{1}{\tilde{\delta }}\left(\frac{e^{-r\mu }}{e-1}\sum _{\nu =0}^{\infty }\frac{{\beta }_{\nu }\left(r\mu \right)}{\nu !}{\int }_0^{\infty }{b}_{r\mathrm{,}\nu }^{\lambda }\left(\theta \right)|\theta -\mu |d\theta \right)\right)\omega (g;\tilde{\delta }).$

In view of Cauchy-Schwarz inequality, one has

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le \{1+\frac{1}{\tilde{\delta }}{\left(\frac{e^{-r\mu }}{e-1}\sum _{\nu =0}^{\infty }\frac{{\beta }_{\nu }\left(r\mu \right)}{\nu !}{\int }_0^{\infty }{b}_{r\mathrm{,}\nu }^{\lambda }\right(\theta \left)d\theta \right)}^{\frac{1}{2}}$

$\times {\left(\frac{e^{-r\mu }}{e-1}\sum _{\nu =0}^{\infty }\frac{{\beta }_{\nu }\left(r\mu \right)}{\nu !}{\int }_0^{\infty }{b}_{r\mathrm{,}\nu }^{\lambda }\left(\theta \right)(\theta -\mu {)}^{2}d\theta \right)}^{\frac{1}{2}}\left\}\omega \right(g;\tilde{\delta })$

$\le \{1+\frac{\sqrt{{\tilde{G}}_{r\mathrm{,}\lambda }(g_2^{\mu };\mu )}}{\tilde{\delta }}\}\omega (g;\tilde{\delta }).$

On taking $\tilde{\delta }={\tilde{G}}_{r\mathrm{,}\lambda }(g_2^{\mu };\mu )$, we yield

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le 2\omega (g;\tilde{\delta }).$

Hence, Theorem (2.2) is proved.

After that, we demonstrate that the provided operators satisfy a theorem of the Voronovskaja type (Özger et al., 2025; Ayman-Mursaleen et al., 2022) in Eq. (4), which provides an asymptotic expansion for twice continuously differentiable functions as:

Theorem 2.3 For $g\mathrm{,}{g}^{\prime }\mathrm{,}{g^{\prime }}^{\prime }\in C\left[\mathrm{0,}\infty \right){\cap }_{}E=\{g:\frac{g\left(\mu \right)}{1+{\mu }^2}$ coverges as $\mu \to \infty \}$ and $\mu \in \left[\mathrm{0,}\infty \right)$, we receive

$\underset{r\to \infty }{\lim }r\left({\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right)=g^{\prime }\left(\mu \right)\left(\frac{\lambda (e-1)+1}{e-1}\right)$

$+\frac{{g^{\prime }}^{\prime }\left(\mu \right)}{2!}\left(2e(\lambda +1)-2\lambda -2\left(\frac{\lambda (e-1)+1}{e-1}\right)\right)\mu \mathrm{.}$

Proof. We have to recall the Taylor series expansion

where $\xi \left(\theta \mathrm{,}\mu \right)$ is denotes the Peano remainder with $\xi \left(\theta \mathrm{,}\mu \right)\in C\left[\mathrm{0,}\infty \right){\cap }_{}E$ such that ${\lim }_{\theta \to \mu }\xi \left(\theta \mathrm{,}\mu \right)=0$. Operating the operators ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ defined by Eq. (4) in the Eq. (10)

Taking limit on both sides in the above expression Eq. (11), we get

Taking into account the Cauchy-Schwarz inequality, the final term evolves into

From Eqs. (12), (13), Lemma (1.3) and ${\lim }_{r\to \infty }{\tilde{G}}_{r\mathrm{,}\lambda }\left({\xi }^2\right(\theta \mathrm{,}\mu );\mu )=0$, we have

$\begin{array}{c}\underset{r\to \infty }{\lim }r\left({\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right)=g^{\prime }\left(\mu \right)\left(\frac{\lambda (e-1)+1}{e-1}\right)\\ +\frac{{g^{\prime }}^{\prime }\left(\mu \right)}{2!}\left(2e(\lambda +1)-2\lambda -2\left(\frac{\lambda (e-1)+1}{e-1}\right)\right)\mu \mathrm{.}\end{array}$

Which proves the desired result.

3. Direct results

Let us define “${\tilde{C}}_{\tilde{B}}\left(\right[\mathrm{0,}\infty \left)\right)$, the space of bounded and continuous functions, and Peetre’s $K$-functional is represented in the following way:

${\tilde{K}}_2\left(g\mathrm{,}\delta \right)=\underset{g\in {\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)}{\inf }\{\parallel g-g{\parallel }_{{\tilde{C}}_{\tilde{B}}\left[\mathrm{0,}\infty \right)}+\tilde{\delta }\parallel {g^{\prime }}^{\prime }{\parallel }_{{\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)}\},$

where ${\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)=\{g\in {\tilde{C}}_{\tilde{B}}[\mathrm{0,}\infty ):g^{\prime }\mathrm{,}{g^{\prime }}^{\prime }\in {\tilde{C}}_{\tilde{B}}[\mathrm{0,}\infty \left)\right\}$ with the norm $\parallel g\parallel =\underset{0\le \mu <\infty }{\sup }\left|g\right(\mu \left)\right|$ and Ditzian-Totik modulus of smoothness of second order is given by

${\tilde{\omega }}_2(g;\sqrt{\tilde{\delta }})=\underset{0<k\le \sqrt{\tilde{\delta }}}{\sup }\underset{\mu \in \left[\mathrm{0,}\infty \right)}{\sup }\left|g\right(\mu +2k)-2g(\mu +k)+g(\mu )|.$

In account of DeVore and Lorentz (Devore and Lorentz, 1993) (see page no. 177, Theorem 2.4) as:

where $\tilde{C}$ represent a absolute constant. Further, proving the local approximation theorems, we consider auxiliary operators as follows:

where $g\in {\tilde{C}}_{\tilde{B}}[\mathrm{0,}\infty ),$ $\mu \ge 0$. Now by using the Eq. (15), one has

Lemma 3.1 Let the operators defined” in Eq. (4) and $\mu \ge 0$, $g\in {\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)$. Then, we have

$\left|{\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le \theta \left(\mu \right)\parallel {g^{\prime }}^{\prime }\parallel \mathrm{,}$

where $\theta \left(\mu \right)={\tilde{G}}_{r\mathrm{,}\lambda }(g_2^{\mu };\mu )+{\left({\tilde{G}}_{r\mathrm{,}\lambda }\right(g_1^{\mu };\mu \left)\right)}^2$.

Proof. For Taylor’s expansion and $g\in {\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)$, one has

By applying the “operators that are designated by ${\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }(.;.)$ and are provided in Eq. (15) to both sides of the equation that is provided in Eq. (17), we obtain:

${\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }(g;\mu )-g\left(\mu \right)=g^{\prime }\left(\mu \right){\tilde{G}}_{r\mathrm{,}\lambda }(g_1^{\mu };\mu )+{\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\left(\underset{\mu }{\overset{\theta }{\int }}\right(\theta -v\left){g^{\prime }}^{\prime }\right(v)dv;\mu ).$

Combining the Eq. (16) and Eq. (17), we have

${\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }(g;\mu )-g\left(\mu \right)={\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\left(\underset{\mu }{\overset{\theta }{\int }}(\theta -v){g^{\prime }}^{\prime }\left(v\right)dv;\mu \right)={\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\left(\underset{\mu }{\overset{\theta }{\int }}(\theta -v){g^{\prime }}^{\prime }\left(v\right)dv;\mu \right)$

$-\underset{\mu }{\overset{\mu +\frac{1}{r}\left(\frac{\lambda (e-1)+1}{e-1}\right)}{\int }}\left(\mu +\frac{1}{r}\left(\frac{\lambda (e-1)+1}{e-1}\right)-v\right){g^{\prime }}^{\prime }\left(v\right)dv\mathrm{,}$

$\left|{\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le \left|{\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\right(\underset{\mu }{\overset{\theta }{\int }}(\theta -v){g^{\prime }}^{\prime }\left(v\right)dv;\mu \left)\right|$

Since,

then”

Considering Eqs. (18), (19) and (20) altogether, we get

$\left|{\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le \left({\tilde{G}}^{*}{}_{r\mathrm{,}\lambda }(g_2^{\mu };\mu )+{\left(\frac{1}{r}\left(\frac{\lambda (e-1)+1}{e-1}\right)\right)}^2\right)\parallel {g^{\prime }}^{\prime }\parallel$

$=\parallel {g^{\prime }}^{\prime }\parallel \theta (\mu ).$

We have therefore reached the desired outcome.

Theorem 3.1 Taking into consideration that $g\in {\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)$ “is present as well as the operators listed in Eq. (4), we are able to establish that

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le \tilde{C}{\tilde{\omega }}_2(g;\sqrt{\theta \left(\mu \right)})+\omega (g;{\tilde{G}}_{r\mathrm{,}\lambda }(g_1^{\mu };\mu )),$

where $\tilde{C}\ge 0$ and $\theta \left(\mu \right)$ is given in Lemma (3.1).

Proof. For $g\in {\tilde{C}}_{\tilde{B}}^2\left[\mathrm{0,}\infty \right)$ and $h\in {\tilde{C}}_{\tilde{B}}\left[\mathrm{0,}\infty \right)$, and on account of ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ given by Eq. (4), one has

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le \left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g-h;\mu \left)\right|+\left|\right(g-h\left)\right(\mu \left)\right|+\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(h;\mu )-h(\mu \left)\right|$

$+\left|g\right({\tilde{G}}_{r\mathrm{,}\lambda }\left(g_1\mathrm{,}\mu \right))-g(\mu )|.$

In view of inequalities in Eq. (16) and Lemma 3.1, one has”

$\begin{array}{c}\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-\mathrm{}g(\mu \left)\right|\le 4\parallel g-\mathrm{}h\parallel +\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(h;\mu )-\mathrm{}h(\mu \left)\right|\\ +|g\left({\tilde{G}}_{r\mathrm{,}\lambda }\left(g_1\mathrm{,}\mu \right)\right)-g(\mu \left)\right|\end{array}$

$\le 4\parallel g-h\parallel +\theta \left(y\right)\parallel {h^{\prime }}^{\prime }\parallel +\omega (g;{\tilde{G}}_{r\mathrm{,}\lambda }((\theta -\mu );\mu )).$

Using Eq. (14), we yield the desired result.

Now, Lipschitz type space (Mansoori et al., 2025; Özarslan and Aktu lu, 2013; Cai et al., 2023), which is

$\begin{array}{c}Li{p}_{\tilde{M}}^{{\phi }_1\mathrm{,}{\phi }_2}\left(\tau \right):=\{g\in {\tilde{C}}_{\tilde{B}}[\mathrm{0,}\infty ):|-g\left(y\right)+g\left(t\right)|\\ \le \tilde{M}\frac{|-\text{ y }+\text{ t}{|}^{\tau }}{{({\phi }_{1}y+{\phi }_2{y}^2+t)}^{\frac{\tau }{2}}}:t\mathrm{,}y\in \left(\mathrm{0,}\infty \right)\},\end{array}$

where $\tilde{M}>0$, $0<\tau \le 1$ and ${\phi }_1\mathrm{,}{\phi }_2>0$.

Theorem 3.2 Let ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ be the operator introduced in Eq. (4). Then, for any $g\in Li{p}_M^{{\phi }_1\mathrm{,}{\phi }_2}\left(\tau \right)$, we have

where the parameters satisfy $0<\tau \le 1$ and ${\phi }_1\mathrm{,}{\phi }_2>0$ and $\lambda \left(r\right)={\tilde{G}}_{r\mathrm{,}\lambda }({\eta }_2;r)$.

Proof. For $r\ge 0$ and $\tau =1$, one yield

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;r)-g(r\left)\right|\le {\tilde{G}}_{r\mathrm{,}\lambda }\left(\right|g\left(t\right)-g\left(r\right)|;r)$

$\le \tilde{M}{\tilde{G}}_{r\mathrm{,}\lambda }(\frac{|t-r|}{{(t+{\phi }_{1}r+{\phi }_2{r}^2)}^{\frac{1}{2}}};r).$

Since $\frac{1}{t+{\phi }_{1}r+{\phi }_2{r}^2}<\frac{1}{{\phi }_{1}r+{\phi }_2{r}^2}$, for $r\in \left(\mathrm{0,}\infty \right)$, we yield

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;r)-g(r\left)\right|\le \frac{\tilde{M}}{{({\phi }_{1}r+{\phi }_2{r}^2)}^{\frac{1}{2}}}{\left({\tilde{G}}_{r\mathrm{,}\lambda }\right({\eta }_2;r\left)\right)}^{\frac{1}{2}}$

$\le \tilde{M}{\left(\frac{\lambda \left(r\right)}{{\phi }_{1}r+{\phi }_2{r}^2}\right)}^{\frac{1}{2}}\mathrm{,}$

his confirms the validity” of Theorem 3.2 for the case $\tau =1$. To extend this result to the remaining parameter range $\tau \in \left(\mathrm{0,1}\right)$, we apply Hölder’s inequality with the conjugate exponents $p=\frac{2}{\tau }$ and $q=\frac{2}{2-\tau }$, yielding

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;r)-g(r\left)\right|\le {\left({\tilde{G}}_{r\mathrm{,}\lambda }\right(\left|g\right(t)-g(r{\left)\right|}^{\frac{2}{\tau }};r\left)\right)}^{\frac{\tau }{2}}$

$\le \tilde{M}{\left({\tilde{G}}_{r\mathrm{,}\lambda }\right(\frac{|t-r{|}^2}{(t+{\phi }_{1}r+{\phi }_2{r}^2)};r\left)\right)}^{\frac{\tau }{2}}\mathrm{.}$

Since $\frac{1}{t+{\phi }_{1}r+{\phi }_2{r}^2}<\frac{1}{{\phi }_{1}r+{\phi }_2{r}^2}$, for $r\in \left(\mathrm{0,}\infty \right)$, one get

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;r)-g(r\left)\right|\le \tilde{M}{\left(\frac{{\tilde{G}}_{r\mathrm{,}\lambda }\left(\right|t-r{|}^2;r)}{{\phi }_{1}r+{\phi }_2{r}^2}\right)}^{\frac{\tau }{2}}\le \tilde{M}{\left(\frac{\lambda \left(r\right)}{{\phi }_{1}r+{\phi }_2{r}^2}\right)}^{\frac{\tau }{2}}\mathrm{.}$

Hence, Theorem 3.2 is proved.

We employ Lenze’s (Lenze 1988) Lipschitz-type maximal function to determine the $b$-th order modulus of continuity, which is used for local approximation analysis:”

In the subsequent step, we investigate the approximation result (locally) in the of the $b^{th}$ order modulus of continuity. The Lipschitz-type maximum function that Lenze (Lenze 1988) provided in his 2015 work is as follows:

Theorem 3.3 For $g\in {\tilde{C}}_{\tilde{B}}\left[\mathrm{0,}\infty \right)$ and $b\in \left(\mathrm{0,1}\right]$, $\forall \mathrm{}\mu \in \left[\mathrm{0,}\infty \right)$, one has

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le {\tilde{\omega }}_r(g;\mu )\left(\lambda \right(\mu {\left)\right)}^{\frac{b}{2}}\mathrm{.}$

Proof. It is found that

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le {\tilde{G}}_{r\mathrm{,}\lambda }\left(\right|g\left(t\right)-g\left(\mu \right)|;\mu ).$

In view of Eq. (23), one has

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le {\tilde{\omega }}_s(g;y){\tilde{G}}_{r\mathrm{,}\lambda }\left(\right|t-\mu {|}^b;\mu ).$

Then, in account of Hölder’s inequality with $p_1=\frac{2}{b}$ and $p_2=\frac{2}{2-b}$, we have

$\left|{\tilde{G}}_{r\mathrm{,}\lambda }\right(g;\mu )-g(\mu \left)\right|\le {\tilde{\omega }}_b(g;\mu )\left({\tilde{G}}_{r\mathrm{,}\lambda }\right(|t-\mu {|}^2;\mu {\left)\right)}^{\frac{b}{2}}\mathrm{.}$

Hence, we prove “the desired result.

4. Bivariate Száz–Gamma-type operators constructed from adjoint Bernoulli polynomials

“This section leverages Adjoint Bernoulli Polynomials to construct and analyze the bivariate Száz-Gamma type operators, ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ which were first presented in Eq. (4). Several researchers have recently investigated multivariate generalizations of various operators (Rao et al., 2025, 2025a,b). Let ${\kappa }^2=\left\{\right({\mu }_1\mathrm{,}{\mu }_2):0\le \mathrm{}{\mu }_1<\infty \mathrm{,0}\le \mathrm{}{\mu }_2<\infty \}$ and $C\left({\kappa }^2\right)$ is set of functions which are continuous on ${\kappa }^2$ with norm

$\parallel g{\parallel }_{C\left({\kappa }^2\right)}=\underset{\left({\mu }_1\mathrm{,}{\mu }_2\right)\in {\kappa }^2}{\sup }\left|g\right({\mu }_1\mathrm{,}{\mu }_2)|.$

Then, for all $g\in C\left({\kappa }^2\right)$ and $r_1\mathrm{,}{r}_2\in \mathbb{N}$. Next, new (bivariant) version of ${\tilde{G}}_{r\mathrm{,}\lambda }(.;.)$ given by Eq. (4) is as follows:

${\tilde{G}}_{r_1\mathrm{,}{r}_2\mathrm{,}\lambda }(g;{\mu }_1\mathrm{,}{\mu }_2)=\sum _{{\nu }_1=0}^{\infty }\sum _{{\nu }_2=0}^{\infty }{g}_{{\nu }_1}\left(r_1{\mu }_1\right)g_{{\nu }_2}\left(r_2{\mu }_2\right)$

where $g_{{\nu }_i}\left(r_i{\mu }_i\right)$ and $b_{r_i\mathrm{,}{\nu }_i}^{\lambda }\left({\theta }_i\right)$ are defined in Eq. (4). Our analysis of the convergence rate and approximation order relies on several key results, which we now present:

In the two-dimensional setting, we define the test functions as” $p_{ij}={\mu }_1^i{\mu }_2^j$ and ${\theta }_{ij}^{{\mu }_1\mathrm{,}{\mu }_2}\left({\theta }_1\mathrm{,}{\theta }_2\right)={\eta }_{ij}\left({\theta }_1\mathrm{,}{\theta }_2\right)=({\theta }_1-{\mu }_1{)}^i{({\theta }_2-{\mu }_2)}^j$, for $i\mathrm{,}j\in \left\{\mathrm{0,1,2}\right\}$ are the test functions and central moments for the bivariate operators respectively.

Lemma 4.1 Let $g\in C\left({\kappa }^2\right)$ and ${\tilde{G}}_{r_1\mathrm{,}{r}_2\mathrm{,}\lambda }(.;.)$ given by Eq. (24) and the test functions $p_{ij}(.;.)$. Then, one has

${\tilde{G}}_{r_1\mathrm{,}{r}_2\mathrm{,}\lambda }(p_{00};{\mu }_1\mathrm{,}{\mu }_2)=\mathrm{1,}$

${\tilde{G}}_{r_1\mathrm{,}{r}_2\mathrm{,}\lambda }(p_{10};{\mu }_1\mathrm{,}{\mu }_2)={\mu }_1+\frac{1}{r_1}\left(\frac{1+\lambda (e-1)}{e-1}\right)\mathrm{,}$

${\tilde{G}}_{r_1\mathrm{,}{r}_2\mathrm{,}\lambda }(p_{01};{\mu }_1\mathrm{,}{\mu }_2)={\mu }_2+\frac{1}{r_2}\left(\frac{\lambda +1(e-1)}{e-1}\right)\mathrm{,}$

$\begin{array}{c}{\tilde{G}}_{r_1\mathrm{,}{r}_2\mathrm{,}\lambda }(p_{20};{\mu }_1\mathrm{,}{\mu }_2)={\mu }_1^2+\frac{{\mu }_1}{r_1}\left(2e\right(\lambda +1)-2\lambda )\\ +\frac{1}{r_1^2}\left(1+\frac{1+2\lambda }{e-1}+{\lambda }^2+\lambda \right)\mathrm{,}\end{array}$